// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H

namespace Eigen {

namespace internal {

template<typename _MatrixType>
struct traits<FullPivHouseholderQR<_MatrixType>> : traits<_MatrixType>
{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	enum
	{
		Flags = 0
	};
};

template<typename MatrixType>
struct FullPivHouseholderQRMatrixQReturnType;

template<typename MatrixType>
struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType>>
{
	typedef typename MatrixType::PlainObject ReturnType;
};

} // end namespace internal

/** \ingroup QR_Module
 *
 * \class FullPivHouseholderQR
 *
 * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the QR decomposition
 *
 * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b P', \b Q and \b R
 * such that
 * \f[
 *  \mathbf{P} \, \mathbf{A} \, \mathbf{P}' = \mathbf{Q} \, \mathbf{R}
 * \f]
 * by using Householder transformations. Here, \b P and \b P' are permutation matrices, \b Q a unitary matrix
 * and \b R an upper triangular matrix.
 *
 * This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
 * numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::fullPivHouseholderQr()
 */
template<typename _MatrixType>
class FullPivHouseholderQR : public SolverBase<FullPivHouseholderQR<_MatrixType>>
{
  public:
	typedef _MatrixType MatrixType;
	typedef SolverBase<FullPivHouseholderQR> Base;
	friend class SolverBase<FullPivHouseholderQR>;

	EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivHouseholderQR)
	enum
	{
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType;
	typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
	typedef Matrix<StorageIndex,
				   1,
				   EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime, RowsAtCompileTime),
				   RowMajor,
				   1,
				   EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime, MaxRowsAtCompileTime)>
		IntDiagSizeVectorType;
	typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
	typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
	typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
	typedef typename MatrixType::PlainObject PlainObject;

	/** \brief Default Constructor.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
	 */
	FullPivHouseholderQR()
		: m_qr()
		, m_hCoeffs()
		, m_rows_transpositions()
		, m_cols_transpositions()
		, m_cols_permutation()
		, m_temp()
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
	}

	/** \brief Default Constructor with memory preallocation
	 *
	 * Like the default constructor but with preallocation of the internal data
	 * according to the specified problem \a size.
	 * \sa FullPivHouseholderQR()
	 */
	FullPivHouseholderQR(Index rows, Index cols)
		: m_qr(rows, cols)
		, m_hCoeffs((std::min)(rows, cols))
		, m_rows_transpositions((std::min)(rows, cols))
		, m_cols_transpositions((std::min)(rows, cols))
		, m_cols_permutation(cols)
		, m_temp(cols)
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
	}

	/** \brief Constructs a QR factorization from a given matrix
	 *
	 * This constructor computes the QR factorization of the matrix \a matrix by calling
	 * the method compute(). It is a short cut for:
	 *
	 * \code
	 * FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
	 * qr.compute(matrix);
	 * \endcode
	 *
	 * \sa compute()
	 */
	template<typename InputType>
	explicit FullPivHouseholderQR(const EigenBase<InputType>& matrix)
		: m_qr(matrix.rows(), matrix.cols())
		, m_hCoeffs((std::min)(matrix.rows(), matrix.cols()))
		, m_rows_transpositions((std::min)(matrix.rows(), matrix.cols()))
		, m_cols_transpositions((std::min)(matrix.rows(), matrix.cols()))
		, m_cols_permutation(matrix.cols())
		, m_temp(matrix.cols())
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
		compute(matrix.derived());
	}

	/** \brief Constructs a QR factorization from a given matrix
	 *
	 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c
	 * MatrixType is a Eigen::Ref.
	 *
	 * \sa FullPivHouseholderQR(const EigenBase&)
	 */
	template<typename InputType>
	explicit FullPivHouseholderQR(EigenBase<InputType>& matrix)
		: m_qr(matrix.derived())
		, m_hCoeffs((std::min)(matrix.rows(), matrix.cols()))
		, m_rows_transpositions((std::min)(matrix.rows(), matrix.cols()))
		, m_cols_transpositions((std::min)(matrix.rows(), matrix.cols()))
		, m_cols_permutation(matrix.cols())
		, m_temp(matrix.cols())
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
		computeInPlace();
	}

#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
	 * \c *this is the QR decomposition.
	 *
	 * \param b the right-hand-side of the equation to solve.
	 *
	 * \returns the exact or least-square solution if the rank is greater or equal to the number of columns of A,
	 * and an arbitrary solution otherwise.
	 *
	 * \note_about_checking_solutions
	 *
	 * \note_about_arbitrary_choice_of_solution
	 *
	 * Example: \include FullPivHouseholderQR_solve.cpp
	 * Output: \verbinclude FullPivHouseholderQR_solve.out
	 */
	template<typename Rhs>
	inline const Solve<FullPivHouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

	/** \returns Expression object representing the matrix Q
	 */
	MatrixQReturnType matrixQ(void) const;

	/** \returns a reference to the matrix where the Householder QR decomposition is stored
	 */
	const MatrixType& matrixQR() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return m_qr;
	}

	template<typename InputType>
	FullPivHouseholderQR& compute(const EigenBase<InputType>& matrix);

	/** \returns a const reference to the column permutation matrix */
	const PermutationType& colsPermutation() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return m_cols_permutation;
	}

	/** \returns a const reference to the vector of indices representing the rows transpositions */
	const IntDiagSizeVectorType& rowsTranspositions() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return m_rows_transpositions;
	}

	/** \returns the absolute value of the determinant of the matrix of which
	 * *this is the QR decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the QR decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \warning a determinant can be very big or small, so for matrices
	 * of large enough dimension, there is a risk of overflow/underflow.
	 * One way to work around that is to use logAbsDeterminant() instead.
	 *
	 * \sa logAbsDeterminant(), MatrixBase::determinant()
	 */
	typename MatrixType::RealScalar absDeterminant() const;

	/** \returns the natural log of the absolute value of the determinant of the matrix of which
	 * *this is the QR decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the QR decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \note This method is useful to work around the risk of overflow/underflow that's inherent
	 * to determinant computation.
	 *
	 * \sa absDeterminant(), MatrixBase::determinant()
	 */
	typename MatrixType::RealScalar logAbsDeterminant() const;

	/** \returns the rank of the matrix of which *this is the QR decomposition.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index rank() const
	{
		using std::abs;
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
		Index result = 0;
		for (Index i = 0; i < m_nonzero_pivots; ++i)
			result += (abs(m_qr.coeff(i, i)) > premultiplied_threshold);
		return result;
	}

	/** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index dimensionOfKernel() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return cols() - rank();
	}

	/** \returns true if the matrix of which *this is the QR decomposition represents an injective
	 *          linear map, i.e. has trivial kernel; false otherwise.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isInjective() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return rank() == cols();
	}

	/** \returns true if the matrix of which *this is the QR decomposition represents a surjective
	 *          linear map; false otherwise.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isSurjective() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return rank() == rows();
	}

	/** \returns true if the matrix of which *this is the QR decomposition is invertible.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isInvertible() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return isInjective() && isSurjective();
	}

	/** \returns the inverse of the matrix of which *this is the QR decomposition.
	 *
	 * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
	 *       Use isInvertible() to first determine whether this matrix is invertible.
	 */
	inline const Inverse<FullPivHouseholderQR> inverse() const
	{
		eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
		return Inverse<FullPivHouseholderQR>(*this);
	}

	inline Index rows() const { return m_qr.rows(); }
	inline Index cols() const { return m_qr.cols(); }

	/** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
	 *
	 * For advanced uses only.
	 */
	const HCoeffsType& hCoeffs() const { return m_hCoeffs; }

	/** Allows to prescribe a threshold to be used by certain methods, such as rank(),
	 * who need to determine when pivots are to be considered nonzero. This is not used for the
	 * QR decomposition itself.
	 *
	 * When it needs to get the threshold value, Eigen calls threshold(). By default, this
	 * uses a formula to automatically determine a reasonable threshold.
	 * Once you have called the present method setThreshold(const RealScalar&),
	 * your value is used instead.
	 *
	 * \param threshold The new value to use as the threshold.
	 *
	 * A pivot will be considered nonzero if its absolute value is strictly greater than
	 *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
	 * where maxpivot is the biggest pivot.
	 *
	 * If you want to come back to the default behavior, call setThreshold(Default_t)
	 */
	FullPivHouseholderQR& setThreshold(const RealScalar& threshold)
	{
		m_usePrescribedThreshold = true;
		m_prescribedThreshold = threshold;
		return *this;
	}

	/** Allows to come back to the default behavior, letting Eigen use its default formula for
	 * determining the threshold.
	 *
	 * You should pass the special object Eigen::Default as parameter here.
	 * \code qr.setThreshold(Eigen::Default); \endcode
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	FullPivHouseholderQR& setThreshold(Default_t)
	{
		m_usePrescribedThreshold = false;
		return *this;
	}

	/** Returns the threshold that will be used by certain methods such as rank().
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	RealScalar threshold() const
	{
		eigen_assert(m_isInitialized || m_usePrescribedThreshold);
		return m_usePrescribedThreshold
				   ? m_prescribedThreshold
				   // this formula comes from experimenting (see "LU precision tuning" thread on the list)
				   // and turns out to be identical to Higham's formula used already in LDLt.
				   : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
	}

	/** \returns the number of nonzero pivots in the QR decomposition.
	 * Here nonzero is meant in the exact sense, not in a fuzzy sense.
	 * So that notion isn't really intrinsically interesting, but it is
	 * still useful when implementing algorithms.
	 *
	 * \sa rank()
	 */
	inline Index nonzeroPivots() const
	{
		eigen_assert(m_isInitialized && "LU is not initialized.");
		return m_nonzero_pivots;
	}

	/** \returns the absolute value of the biggest pivot, i.e. the biggest
	 *          diagonal coefficient of U.
	 */
	RealScalar maxPivot() const { return m_maxpivot; }

#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	void _solve_impl(const RhsType& rhs, DstType& dst) const;

	template<bool Conjugate, typename RhsType, typename DstType>
	void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

  protected:
	static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

	void computeInPlace();

	MatrixType m_qr;
	HCoeffsType m_hCoeffs;
	IntDiagSizeVectorType m_rows_transpositions;
	IntDiagSizeVectorType m_cols_transpositions;
	PermutationType m_cols_permutation;
	RowVectorType m_temp;
	bool m_isInitialized, m_usePrescribedThreshold;
	RealScalar m_prescribedThreshold, m_maxpivot;
	Index m_nonzero_pivots;
	RealScalar m_precision;
	Index m_det_pq;
};

template<typename MatrixType>
typename MatrixType::RealScalar
FullPivHouseholderQR<MatrixType>::absDeterminant() const
{
	using std::abs;
	eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
	eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
	return abs(m_qr.diagonal().prod());
}

template<typename MatrixType>
typename MatrixType::RealScalar
FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
{
	eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
	eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
	return m_qr.diagonal().cwiseAbs().array().log().sum();
}

/** Performs the QR factorization of the given matrix \a matrix. The result of
 * the factorization is stored into \c *this, and a reference to \c *this
 * is returned.
 *
 * \sa class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)
 */
template<typename MatrixType>
template<typename InputType>
FullPivHouseholderQR<MatrixType>&
FullPivHouseholderQR<MatrixType>::compute(const EigenBase<InputType>& matrix)
{
	m_qr = matrix.derived();
	computeInPlace();
	return *this;
}

template<typename MatrixType>
void
FullPivHouseholderQR<MatrixType>::computeInPlace()
{
	check_template_parameters();

	using std::abs;
	Index rows = m_qr.rows();
	Index cols = m_qr.cols();
	Index size = (std::min)(rows, cols);

	m_hCoeffs.resize(size);

	m_temp.resize(cols);

	m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);

	m_rows_transpositions.resize(size);
	m_cols_transpositions.resize(size);
	Index number_of_transpositions = 0;

	RealScalar biggest(0);

	m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
	m_maxpivot = RealScalar(0);

	for (Index k = 0; k < size; ++k) {
		Index row_of_biggest_in_corner, col_of_biggest_in_corner;
		typedef internal::scalar_score_coeff_op<Scalar> Scoring;
		typedef typename Scoring::result_type Score;

		Score score = m_qr.bottomRightCorner(rows - k, cols - k)
						  .unaryExpr(Scoring())
						  .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
		row_of_biggest_in_corner += k;
		col_of_biggest_in_corner += k;
		RealScalar biggest_in_corner =
			internal::abs_knowing_score<Scalar>()(m_qr(row_of_biggest_in_corner, col_of_biggest_in_corner), score);
		if (k == 0)
			biggest = biggest_in_corner;

		// if the corner is negligible, then we have less than full rank, and we can finish early
		if (internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision)) {
			m_nonzero_pivots = k;
			for (Index i = k; i < size; i++) {
				m_rows_transpositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
				m_cols_transpositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
				m_hCoeffs.coeffRef(i) = Scalar(0);
			}
			break;
		}

		m_rows_transpositions.coeffRef(k) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner);
		m_cols_transpositions.coeffRef(k) = internal::convert_index<StorageIndex>(col_of_biggest_in_corner);
		if (k != row_of_biggest_in_corner) {
			m_qr.row(k).tail(cols - k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols - k));
			++number_of_transpositions;
		}
		if (k != col_of_biggest_in_corner) {
			m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
			++number_of_transpositions;
		}

		RealScalar beta;
		m_qr.col(k).tail(rows - k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
		m_qr.coeffRef(k, k) = beta;

		// remember the maximum absolute value of diagonal coefficients
		if (abs(beta) > m_maxpivot)
			m_maxpivot = abs(beta);

		m_qr.bottomRightCorner(rows - k, cols - k - 1)
			.applyHouseholderOnTheLeft(m_qr.col(k).tail(rows - k - 1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k + 1));
	}

	m_cols_permutation.setIdentity(cols);
	for (Index k = 0; k < size; ++k)
		m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));

	m_det_pq = (number_of_transpositions % 2) ? -1 : 1;
	m_isInitialized = true;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename _MatrixType>
template<typename RhsType, typename DstType>
void
FullPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
	const Index l_rank = rank();

	// FIXME introduce nonzeroPivots() and use it here. and more generally,
	// make the same improvements in this dec as in FullPivLU.
	if (l_rank == 0) {
		dst.setZero();
		return;
	}

	typename RhsType::PlainObject c(rhs);

	Matrix<typename RhsType::Scalar, 1, RhsType::ColsAtCompileTime> temp(rhs.cols());
	for (Index k = 0; k < l_rank; ++k) {
		Index remainingSize = rows() - k;
		c.row(k).swap(c.row(m_rows_transpositions.coeff(k)));
		c.bottomRightCorner(remainingSize, rhs.cols())
			.applyHouseholderOnTheLeft(m_qr.col(k).tail(remainingSize - 1), m_hCoeffs.coeff(k), &temp.coeffRef(0));
	}

	m_qr.topLeftCorner(l_rank, l_rank).template triangularView<Upper>().solveInPlace(c.topRows(l_rank));

	for (Index i = 0; i < l_rank; ++i)
		dst.row(m_cols_permutation.indices().coeff(i)) = c.row(i);
	for (Index i = l_rank; i < cols(); ++i)
		dst.row(m_cols_permutation.indices().coeff(i)).setZero();
}

template<typename _MatrixType>
template<bool Conjugate, typename RhsType, typename DstType>
void
FullPivHouseholderQR<_MatrixType>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
	const Index l_rank = rank();

	if (l_rank == 0) {
		dst.setZero();
		return;
	}

	typename RhsType::PlainObject c(m_cols_permutation.transpose() * rhs);

	m_qr.topLeftCorner(l_rank, l_rank)
		.template triangularView<Upper>()
		.transpose()
		.template conjugateIf<Conjugate>()
		.solveInPlace(c.topRows(l_rank));

	dst.topRows(l_rank) = c.topRows(l_rank);
	dst.bottomRows(rows() - l_rank).setZero();

	Matrix<Scalar, 1, DstType::ColsAtCompileTime> temp(dst.cols());
	const Index size = (std::min)(rows(), cols());
	for (Index k = size - 1; k >= 0; --k) {
		Index remainingSize = rows() - k;

		dst.bottomRightCorner(remainingSize, dst.cols())
			.applyHouseholderOnTheLeft(m_qr.col(k).tail(remainingSize - 1).template conjugateIf<!Conjugate>(),
									   m_hCoeffs.template conjugateIf<Conjugate>().coeff(k),
									   &temp.coeffRef(0));

		dst.row(k).swap(dst.row(m_rows_transpositions.coeff(k)));
	}
}
#endif

namespace internal {

template<typename DstXprType, typename MatrixType>
struct Assignment<DstXprType,
				  Inverse<FullPivHouseholderQR<MatrixType>>,
				  internal::assign_op<typename DstXprType::Scalar, typename FullPivHouseholderQR<MatrixType>::Scalar>,
				  Dense2Dense>
{
	typedef FullPivHouseholderQR<MatrixType> QrType;
	typedef Inverse<QrType> SrcXprType;
	static void run(DstXprType& dst,
					const SrcXprType& src,
					const internal::assign_op<typename DstXprType::Scalar, typename QrType::Scalar>&)
	{
		dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
	}
};

/** \ingroup QR_Module
 *
 * \brief Expression type for return value of FullPivHouseholderQR::matrixQ()
 *
 * \tparam MatrixType type of underlying dense matrix
 */
template<typename MatrixType>
struct FullPivHouseholderQRMatrixQReturnType : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType>>
{
  public:
	typedef typename FullPivHouseholderQR<MatrixType>::IntDiagSizeVectorType IntDiagSizeVectorType;
	typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
	typedef Matrix<typename MatrixType::Scalar,
				   1,
				   MatrixType::RowsAtCompileTime,
				   RowMajor,
				   1,
				   MatrixType::MaxRowsAtCompileTime>
		WorkVectorType;

	FullPivHouseholderQRMatrixQReturnType(const MatrixType& qr,
										  const HCoeffsType& hCoeffs,
										  const IntDiagSizeVectorType& rowsTranspositions)
		: m_qr(qr)
		, m_hCoeffs(hCoeffs)
		, m_rowsTranspositions(rowsTranspositions)
	{
	}

	template<typename ResultType>
	void evalTo(ResultType& result) const
	{
		const Index rows = m_qr.rows();
		WorkVectorType workspace(rows);
		evalTo(result, workspace);
	}

	template<typename ResultType>
	void evalTo(ResultType& result, WorkVectorType& workspace) const
	{
		using numext::conj;
		// compute the product H'_0 H'_1 ... H'_n-1,
		// where H_k is the k-th Householder transformation I - h_k v_k v_k'
		// and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
		const Index rows = m_qr.rows();
		const Index cols = m_qr.cols();
		const Index size = (std::min)(rows, cols);
		workspace.resize(rows);
		result.setIdentity(rows, rows);
		for (Index k = size - 1; k >= 0; k--) {
			result.block(k, k, rows - k, rows - k)
				.applyHouseholderOnTheLeft(
					m_qr.col(k).tail(rows - k - 1), conj(m_hCoeffs.coeff(k)), &workspace.coeffRef(k));
			result.row(k).swap(result.row(m_rowsTranspositions.coeff(k)));
		}
	}

	Index rows() const { return m_qr.rows(); }
	Index cols() const { return m_qr.rows(); }

  protected:
	typename MatrixType::Nested m_qr;
	typename HCoeffsType::Nested m_hCoeffs;
	typename IntDiagSizeVectorType::Nested m_rowsTranspositions;
};

// template<typename MatrixType>
// struct evaluator<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
//  : public evaluator<ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> > >
// {};

} // end namespace internal

template<typename MatrixType>
inline typename FullPivHouseholderQR<MatrixType>::MatrixQReturnType
FullPivHouseholderQR<MatrixType>::matrixQ() const
{
	eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
	return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
}

/** \return the full-pivoting Householder QR decomposition of \c *this.
 *
 * \sa class FullPivHouseholderQR
 */
template<typename Derived>
const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::fullPivHouseholderQr() const
{
	return FullPivHouseholderQR<PlainObject>(eval());
}

} // end namespace Eigen

#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
